Even Discrete Cosine Transform The Chinese University of Hong Kong MMAT 5390: Mathematical Imaging Lecture 6: More about DFT & Even Discrete Cosine Transform Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong

Frequency spectrum of an image

Frequency spectrum of an image

Image Enhancement Linear filtering: Modifying a pixel value (in the spatial domain) by a linear combination of neighborhood values. Operations in spatial domain v.s. operations in frequency domains: Linear filtering (matrix multiplication in spatial domain) = discrete convolution In the frequency domain, it is equivalent to multiplying the Fourier transform of the image with a certain function that “kills” or modifies certain frequency components

Spatial transform v.s. frequency transform Discrete convolution: (Matrix multiplication, which define output value as linear combination of its neighborhood) DFT of Discrete convolution: Product of fourier transform DFT(convolution of f and w) = C*DFT(f)*DFT(w) Multiplying the Fourier transform of the image with a certain function that “kills” or modifies certain frequency components

Image components LP = Low Pass; HP = High Pass

Image components

Gaussian noise Example of Gaussian noises:

White noise Example of white noises:

White noise Example of white noises:

Noises as high frequency component Why noises are often considered as high frequency component? (a) Clean image spectrum and Noise spectrum (Noise dominates the high-frequency component); (b) Filtering of high-frequency component